Problem: All of the 4th grade teachers and students from Springer went on a field trip to an art museum. Tickets were $$7.50$ each for teachers and $$2.00$ each for students, and the group paid $$44.00$ in total. A few weeks later, the same group visited a science museum where the tickets cost $$30.00$ each for teachers and $$11.50$ each for students, and the group paid $$200.50$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7.5x+2y = 44}$ ${30x+11.5y = 200.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-30x-8y = -176}$ ${30x+11.5y = 200.5}$ Add the top and bottom equations together. $ 3.5y = 24.5 $ $ y = \dfrac{24.5}{3.5}$ ${y = 7}$ Now that you know ${y = 7}$ , plug it back into $ {7.5x+2y = 44}$ to find $x$ ${7.5x + 2}{(7)}{= 44}$ $7.5x+14 = 44$ $7.5x = 30$ $x = \dfrac{30}{7.5}$ ${x = 4}$ You can also plug ${y = 7}$ into $ {30x+11.5y = 200.5}$ and get the same answer for $x$ ${30x + 11.5}{(7)}{= 200.5}$ ${x = 4}$ There were $4$ teachers and $7$ students on the field trips.